Math, Set 7, Rounds 5-8

13) MATH Short Answer Mother Goose divided her estate among her four favorite friends according to the following specifications: • Humpty Dumpty shall receive half of the estate, less $9000; • Simple Simon shall receive one third of the estate, less $3000; • Little Jack Horner shall receive exactly one fourth of the estate; • Little Miss Muffet shall receive one sixth of the estate, plus $3000. What was the total dollar value of Mother Goose's estate?

ANSWER: $36,000

5) MATH Short Answer What are the coordinates of the points of intersection of the graphs of y = 2x^2 - x and y = 2x + 35?

ANSWER: (-7/2, 28) AND (5, 45) (ACCEPT: (-3.5, 28) AND (5, 45))

20) MATH Short Answer A graph has symmetry with respect to the origin and passes through the point (-2, 3). What other point must be on this graph?

ANSWER: (2, -3)

16) MATH Short Answer The probability that a doctor is correct when making a diagnosis is 0.8. If the doctor makes an incorrect diagnosis, the probability that a patient sues is 0.3. What is the probability that the doctor makes an incorrect diagnosis and the patient sues?

ANSWER: 0.06 (ACCEPT: 3/50 or 6%)

14) MATH Short Answer A card is drawn at random from a standard deck of 52 cards. Given that the card drawn is a face card, what is the probability that the drawn card is a Jack?

ANSWER: 1/3

10) MATH Short Answer Evaluate: ∫ 4 0 4 dx [the integral from 0 to 4 of 4 dx].

ANSWER: 16

21) MATH Short Answer The IQ of a Martian varies directly with the square of the number of tentacles it has. If a Martian with 5 tentacles has an IQ of 75, what is the IQ of a Martian having 8 tentacles?

ANSWER: 192

2) MATH Short Answer Given 2 two by two matrices A = [1 2 3 1] [row one: 1, 3; row 2: 2, 1] and B = [3 1 4 0] [row 1: 3, 4; row 2: 1, 0]. Find the determinant of the product AB.

ANSWER: 20

3) MATH Short Answer A cube painted red on all sides has an edge length of 4 inches. If the cube is then cut into 64 cubes of edge length 1 inch, how many of the 1 inch cubes have exactly one face painted red?

ANSWER: 24

7) MATH Short Answer If f(x) = 2 - x [f of x equals 2 minus x], find f(f '(x)) [f of f prime of x].

ANSWER: 3

20) MATH Short Answer A line in parametric mode is given by the equations: x = 1/2t + 2 and y = 2t. What is the slope of this line?

ANSWER: 4

1) MATH Short Answer A point moves on the curve 6x^2 - 3y^2 = 18 so that its y-coordinate increases at a constant rate of 12 meters per second. At what rate, in meters per second, is the x-coordinate changing when the point is at (3, 2√3 )?

ANSWER: 4√3

16) MATH Short Answer A drawer contains 30 socks: 10 brown, 9 black, 6 blue, and 5 red. What is the fewest number of socks I must pick from the drawer without looking to be absolutely certain I have 2 socks of the same color?

ANSWER: 5

10) MATH Short Answer A farmer has three hens of varying weights. When the hens are weighed in pairs, they weigh 11.3, 11.8, and 12.1 pounds. What is the weight in pounds of the lightest hen?

ANSWER: 5.5

23) MATH Short Answer Given that 𝑓(𝑥) = (2𝑥)/𝑥−1 [f of x equals the fraction with numerator 2x and denominator x minus 1] is a one-to-one function, find x so that f^-1(x) = 5 [f inverse of x equals 5].

ANSWER: 5/2 (ACCEPT: 2.5 OR 2 1/2)

19) MATH Short Answer If each interior angle of a polygon is 14 times the size of each exterior angle, what is the sum in degrees of the interior angles of this polygon?

ANSWER: 5040°

14) MATH Short Answer What is the next number in the arithmetic sequence that has first three terms of 17, 31, and 45?

ANSWER: 59

8) MATH Short Answer How many subsets are there of a set whose elements are the composite numbers less than 9?

ANSWER: 8

21) MATH Multiple Choice If y = (2cos)^(2)x [y equals 2 cosine squared of x], what is dy/dx at x = 𝜋/4? W) -2 X) -1 Y) 1 Z) 2

ANSWER: W) -2

19) MATH Multiple Choice If X and Y are bounded sets, then which of the following is true? W) Both the union and the intersection of sets X and Y are bounded X) Both the union and the intersection of sets X and Y are unbounded Y) The union of sets X and Y is bounded and the intersection of sets X and Y is unbounded Z) The union of sets X and Y is unbounded and the intersection of sets X and Y is bounded

ANSWER: W) BOTH THE UNION AND THE INTERSECTION OF SETS X AND Y ARE BOUNDED

13) MATH Multiple Choice Given 𝑓(𝑥) = √(7𝑥^2 − 3 [f of x equals the square root of the quantity 7 x squared minus 3], which of the following are u(x) and g(u) such that f(x) = g(u(x)) [f of x equals g of u of x]? W) u(x) = 7x^2 − 3 and 𝑔(𝑢) = √𝑢 X) u(x) = √𝑥 and g(u) = 7u^2 - 3 Y) u(x) = √(7𝑥^2 − 3) [u of x equals the square root of the quantity 7 x squared minus 3] and 𝑔(𝑢) = √𝑢 Z) u(x) = √7(𝑥 − 3) and g(u) = 𝑢^2

ANSWER: W) u(x) = 7x^2 − 3 AND 𝑔(𝑢) = √u

8) MATH Multiple Choice A balloon is being filled with water. Its volume in gallons is given by the function V(r) = (r)^3/10 [V of r equals r cubed over 10], where r is the radius of the balloon in inches. What is the average rate of change in gallons per inch of the volume of the balloon as the radius is increased from 1 to 2 inches? W) 0.3 X) 0.7 Y) 0.8 Z) 1.2

ANSWER: X) 0.7

1) MATH Multiple Choice A positive three digit integer n is chosen at random, with each three digit number having an equal chance of being chosen. What is the probability that log2n [log base 2 of n] is an integer? W) 3/899 X) 1/300 Y) 1/222 Z) 1/150

ANSWER: X) 1/300

9) MATH Multiple Choice The price of an item is discounted by 10% and then the new price is further discounted by 20%. What single discount is equivalent to this double discount? W) 15% X) 28% Y) 30% Z) 72%

ANSWER: X) 28%

15) MATH Multiple Choice In the mid-state football conference, each team plays every other team exactly once during a calendar year. If a total of 45 conference games were played in 2012, how many teams were in the conference? W) 8 X) 9 Y) 10 Z) 11

ANSWER: Y) 10

3) MATH Multiple Choice A point has rectangular coordinates (3, 3). What are its equivalent polar coordinates? W) (3, 𝜋/6) X) (3, 𝜋/4) Y) (3√2, 𝜋/6) Z) (3√2, 𝜋/4)

ANSWER: Z) (3√2, 𝜋/4)

7) MATH Short Answer Express the following in terms of a single trigonometric function, given that all fractions are defined: (secx + cscx)/(1 + tanx) [the fraction with numerator secant of x plus cosecant of x and denominator 1 plus the tangent of x].

ANSWER: cscx (ACCEPT: 1/sinx)

5) MATH Short Answer Given that t ≠ 0, find 𝑓( 1 /𝑡 ) if f (t ) =1/t.

ANSWER: t

9) MATH Short Answer Let g(x) = (x − 1)^2 [g of x equal the square of the quantity x minus 1]. What is the set of all real values of x for which g(x) > 4 [g of x is greater than 4]?

ANSWER: x < -1 OR x > 3 (MUST GIVE BOTH ANSWERS)

15) MATH Short Answer Given 𝑓(𝑥) = √(2 − 𝑥) [f of x equals the square root of the quantity 2 minus x], for what value or values of x does f(x+2) = 1 [f of the quantity x plus 2 equals 1]?

ANSWER: x = -1

2) MATH Short Answer If f(x) = x^2 - 1 and g(x) = x - 1, what is the composite function f(g(x)) [f of g of x]?

ANSWER: x^2 - 2x

23) MATH Short Answer Providing your answer in slope intercept form, what is the equation of the normal line to the function f(x) = x^3 + 2x^2 + 5x - 2 at the point where x = 0?

ANSWER: y = (-1/5)x - 2